What is a PRP ?
A PRP is a probable prime number, a number that nobody knows how to prove
or disprove its primality.
Here are some criterions that can be used to recognize these numbers
:
- N does not have any prime factor up to 2^32.
- N cannot be trivially written into a product.
- Take 3 square-free consecutive bases such that (b1/N)=(b2/N)=(b3/N)= -1. N
should be such that b1^((N-1)/2)=b2^((N-1)/2)=b3^((N-1)/2)= -1 (mod N).
- Take a prime base such that (b/N)= -1 and b<>b1<>b2<>b3.
N should pass the strong test in base b.
- or more (if someone has got an idea) !!!
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Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
9751 | (796*10^97183-31)/9 | 97185 | Robert Price | 07/2022 |
9752 | (2^322789+2214)/2 | 97169 | Stefano Morozzi | 09/2020 |
9753 | 2^322725+75761 | 97150 | Enzo Doria | 04/2016 |
9754 | (13*2^322686+1)/119 | 97138 | Lelio R Paula | 11/2013 |
9755 | (2175^29101-1)/2174 | 97121 | Serge Batalov | 06/2022 |
9756 | 22402^21639+21639^22402 | 97118 | Hans Havermann | 07/2020 |
9757 | 23399^14128+14128^23399 | 97108 | Hans Havermann | 06/2020 |
9758 | (2892^28057-1)/2891 | 97108 | Serge Batalov | 06/2022 |
9759 | (9578^24391-1)/9577 | 97104 | Serge Batalov | 06/2022 |
9760 | (2^322525+2606)/2 | 97090 | Stefano Morozzi | 09/2020 |
9761 | (9563^24391-1)/9562 | 97087 | Serge Batalov | 06/2022 |
9762 | (41*10^97085+373)/9 | 97086 | Robert Price | 04/2017 |
9763 | (2^322473+854)/2 | 97074 | Stefano Morozzi | 02/2021 |
9764 | 2^322413+77 | 97056 | Lelio R Paula | 12/2012 |
9765 | 2^322406+177 | 97054 | Lelio R Paula | 12/2012 |
9766 | (349^38167-1)/348 | 97050 | Serge Batalov | 06/2022 |
9767 | (14*10^97044+31)/9 | 97045 | Robert Price | 02/2015 |
9768 | 2^322302+255255 | 97023 | Lelio R Paula | 07/2016 |
9769 | 2^322286+212659 | 97018 | Enzo Doria | 09/2023 |
9770 | (1302^31147-1)/1301 | 97008 | Serge Batalov | 06/2022 |
9771 | 2^322196-113093 | 96991 | Enzo Doria | 07/2014 |
9772 | (2^322169+2690)/2 | 96983 | Stefano Morozzi | 09/2020 |
9773 | (49*10^96980+221)/9 | 96981 | Robert Price | 02/2022 |
9774 | 9*10^96968-47 | 96969 | Robert Price | 11/2021 |
9775 | (5895^25717-1)/5894 | 96962 | Serge Batalov | 06/2022 |
9776 | (27473^21841-1)/27472 | 96946 | Serge Batalov | 06/2022 |
9777 | 23109^15668+15668^23109 | 96943 | Hans Havermann | 07/2020 |
9778 | 2^322000-1861469 | 96932 | Enzo Doria | 03/2016 |
9779 | 2^321971-112747 | 96923 | Enzo Doria | 06/2014 |
9780 | 23754^12023+12023^23754 | 96917 | Hans Havermann | 07/2020 |
9781 | 11^93062-92 | 96915 | Lelio R Paula | 06/2016 |
9782 | (37871^21169-1)/37870 | 96914 | Serge Batalov | 06/2022 |
9783 | (2^321922+2214)/2 | 96908 | Stefano Morozzi | 09/2020 |
9784 | (2^321900+2086)/2 | 96902 | Stefano Morozzi | 09/2020 |
9785 | (11*2^321841-7)/15 | 96884 | Renaud Lifchitz | 03/2011 |
9786 | 3^203051+34 | 96880 | Lelio R Paula | 05/2016 |
9787 | 2*10^96867-51 | 96868 | Robert Price | 11/2017 |
9788 | (2^321779+2618)/2 | 96865 | Stefano Morozzi | 09/2020 |
9789 | (64*10^96863-1)/27 | 96864 | Serge Batalov | 05/2010 |
9790 | 2^321716-2786939 | 96847 | Enzo Doria | 03/2016 |
9791 | 22496^20181+20181^22496 | 96844 | Hans Havermann | 03/2020 |
9792 | 2^321701-113053 | 96842 | Enzo Doria | 07/2014 |
9793 | 2^321700-28635 | 96842 | Lelio R Paula | 06/2016 |
9794 | (5*2^321691-1)/27 | 96838 | Lelio R Paula | 02/2014 |
9795 | 6*10^96835+13 | 96836 | Robert Price | 10/2021 |
9796 | 6^124433-124433 | 96828 | Robert Price | 01/2014 |
9797 | 2^321631+69 | 96821 | Lelio R Paula | 12/2012 |
9798 | (27109^21841-1)/27108 | 96820 | Serge Batalov | 06/2022 |
9799 | (13*2^321606+1)/119 | 96813 | Lelio R Paula | 11/2013 |
9800 | (5052^26141-1)/5051 | 96809 | Serge Batalov | 06/2022 |
9801 | (46*10^96798-1)/9 | 96799 | Serge Batalov | 10/2014 |
9802 | (59*10^96798+31)/9 | 96799 | Robert Price | 09/2015 |
9803 | (2^321536+2062)/2 | 96792 | Stefano Morozzi | 09/2020 |
9804 | 223319#+231223 | 96790 | Makoto Morimoto | 02/2016 |
9805 | 2^321519-28635 | 96787 | Lelio R Paula | 06/2016 |
9806 | 2^321487-1867237 | 96778 | Enzo Doria | 03/2016 |
9807 | 2^321461-63 | 96770 | Lelio R Paula | 12/2012 |
9808 | (16053^23011-1)/16052 | 96770 | Serge Batalov | 06/2022 |
9809 | 2^321439+123 | 96763 | Lelio R Paula | 12/2012 |
9810 | 2^321434+709 | 96762 | Lelio R Paula | 02/2013 |
9811 | 2^321419+22703 | 96757 | Enzo Doria | 09/2023 |
9812 | (2^321415+554)/2 | 96756 | Stefano Morozzi | 09/2020 |
9813 | 14^84413+89 | 96749 | Lelio R Paula | 07/2016 |
9814 | (28*10^96743+71)/9 | 96744 | Serge Batalov | 12/2008 |
9815 | 13^86847+96 | 96743 | Lelio R Paula | 06/2016 |
9816 | 9*2^321336-7 | 96733 | Lelio R Paula | 08/2014 |
9817 | (209*10^96722-17)/3 | 96724 | Robert Price | 05/2017 |
9818 | (54163^20431-1)/54162 | 96710 | Serge Batalov | 06/2022 |
9819 | 2^321237-4615 | 96702 | Yves Minet | 05/2019 |
9820 | (27273^21799-1)/27272 | 96691 | Serge Batalov | 06/2022 |
9821 | 2^321187-1159 | 96687 | Yves Minet | 11/2012 |
9822 | 2^321182-1850081 | 96686 | Enzo Doria | 08/2015 |
9824 | 23359^13768+13768^23359 | 96680 | Hans Havermann | 06/2020 |
9823 | (65630^20071-1)/65629 | 96680 | Serge Batalov | 06/2022 |
9825 | 2^321116-112829 | 96666 | Enzo Doria | 06/2014 |
9826 | 2^321108-109499 | 96664 | Enzo Doria | 04/2014 |
9827 | 2^321098+2793859 | 96661 | Enzo Doria | 03/2016 |
9828 | (2^321100+1170)/2 | 96661 | Stefano Morozzi | 09/2020 |
9829 | 2^321057+4097 | 96648 | Lelio R Paula | 06/2016 |
9830 | (4435*29^66081-4)/7 | 96640 | Curtis Bright | 05/2015 |
9831 | 22743^17750+17750^22743 | 96640 | Hans Havermann | 04/2020 |
9832 | 22882^16679+16679^22882 | 96612 | Hans Havermann | 02/2020 |
9833 | 23452^13163+13163^23452 | 96608 | Hans Havermann | 02/2020 |
9834 | (1042^32009-1)/1041 | 96596 | Serge Batalov | 06/2022 |
9835 | (85*10^96594-373)/9 | 96595 | Robert Price | 05/2021 |
9836 | (2^320863+666)/2 | 96590 | Stefano Morozzi | 09/2020 |
9837 | p(19849)#/2-64 | 96581 | Robert Price | 04/2017 |
9838 | 2^320817+189 | 96576 | Lelio R Paula | 12/2012 |
9839 | 2^320816-4685 | 96576 | Yves Minet | 05/2019 |
9841 | (361*10^96567-163)/9 | 96569 | Robert Price | 10/2017 |
9840 | (19*10^96568+137)/3 | 96569 | Robert Price | 03/2022 |
9842 | 2^320788-110327 | 96567 | Enzo Doria | 05/2014 |
9843 | (3065^27697-1)/3064 | 96561 | Serge Batalov | 06/2022 |
9844 | 2^320764+793 | 96560 | Lelio R Paula | 02/2013 |
9845 | (2^320752+1842)/2 | 96556 | Stefano Morozzi | 02/2021 |
9846 | (48331^20611-1)/48330 | 96542 | Serge Batalov | 06/2022 |
9847 | (107746^19321-1)/(107746^139-1) | 96532 | Ryan Propper | 11/2023 |
9848 | (50595^20521-1)/50594 | 96529 | Serge Batalov | 06/2022 |
9849 | (9179*2^320652-1)/221 | 96528 | Lelio R Paula | 05/2015 |
9850 | 23288^13947+13947^23288 | 96517 | Hans Havermann | 07/2020 |
9852 | 2^320616-78795 | 96516 | Lelio R Paula | 03/2016 |
9851 | (19*10^96520+287)/9/109/409 | 96516 | Alfred Reich | 11/2022 |
9853 | 2^320612-461939 | 96514 | Enzo Doria | 07/2015 |
9854 | (21245^22303-1)/21244 | 96507 | Serge Batalov | 06/2022 |
9855 | 10^96504+987654321 | 96505 | Robert Price | 12/2019 |
9856 | 10^96504+71887 | 96505 | Patrick De Geest | 02/2020 |
9857 | 22654^18195+18195^22654 | 96505 | Hans Havermann | 07/2020 |
9858 | 10^96504-139277 | 96504 | Patrick De Geest | 02/2020 |
9859 | (218^41263-1)/217 | 96490 | Serge Batalov | 06/2022 |
9860 | (47220^20641-1)/47219 | 96474 | Serge Batalov | 06/2022 |
9861 | 5*2^320431+7 | 96461 | Lelio R Paula | 08/2014 |
9862 | 19^75431-90 | 96458 | Lelio R Paula | 06/2016 |
9863 | (11853^23677-1)/11852 | 96452 | Serge Batalov | 06/2022 |
9864 | (2^320386+2814)/2 | 96446 | Stefano Morozzi | 02/2021 |
9865 | (2^320370+3)/67 | 96440 | Lelio R Paula | 10/2013 |
9866 | 3*10^96431-17 | 96432 | Robert Price | 03/2021 |
9867 | 2^320309+1871477 | 96423 | Enzo Doria | 05/2016 |
9868 | (10412^24001-1)/10411 | 96421 | Serge Batalov | 06/2022 |
9869 | tau(809^6028) | 96410 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
9870 | (29*10^96407-23)/3 | 96408 | Robert Price | 03/2022 |
9871 | (21*2^320252-1)/1895 | 96404 | Lelio R Paula | 02/2014 |
9872 | 2^320232-173 | 96400 | Lelio R Paula | 12/2012 |
9873 | 175^42975-2 | 96395 | Lelio R Paula | 10/2017 |
9874 | (2^320170+2718)/2 | 96381 | Stefano Morozzi | 09/2020 |
9875 | 10^96377+199911 | 96378 | Patrick De Geest | 12/2016 |
9876 | 10^96377-20001 | 96377 | Robert Price | 11/2016 |
9877 | (18079^22639-1)/18078 | 96374 | Serge Batalov | 06/2022 |
9879 | 2^320124-2790449 | 96367 | Enzo Doria | 03/2016 |
9878 | 10^96366+19197 | 96367 | Patrick De Geest | 08/2021 |
9881 | 10^96366-10000000001 | 96366 | Robert Price | 04/2021 |
9880 | 10^96366-287367 | 96366 | Patrick De Geest | 08/2021 |
9882 | (26351^21799-1)/26350 | 96365 | Serge Batalov | 06/2022 |
9883 | 22488^19255+19255^22488 | 96351 | Hans Havermann | 11/2019 |
9884 | 23248^13939+13939^23248 | 96346 | Hans Havermann | 11/2019 |
9885 | (5928^25537-1)/5927 | 96345 | Serge Batalov | 06/2022 |
9887 | 2^320043-108775 | 96343 | Enzo Doria | 04/2014 |
9886 | 24989^7168+7168^24989 | 96343 | Hans Havermann | 11/2019 |
9888 | 10^96338+424939 | 96339 | Patrick De Geest | 03/2016 |
9889 | 10^96338-9281 | 96338 | Robert Price | 09/2015 |
9890 | (8527^24509-1)/8526 | 96336 | Serge Batalov | 06/2022 |
9891 | (3*2^320008+1)/7 | 96332 | Lelio R Paula | 09/2013 |
9892 | (2444^28433-1)/2443 | 96331 | Serge Batalov | 06/2022 |
9893 | (48690^20551-1)/48689 | 96327 | Serge Batalov | 06/2022 |
9894 | (61935^20101-1)/61934 | 96318 | Serge Batalov | 06/2022 |
9895 | 25699^5596+5596^25699 | 96317 | Hans Havermann | 07/2020 |
9896 | (7*10^96312+83)/9 | 96312 | Tyler Busby | 01/2023 |
9897 | (11*10^96298+529)/9 | 96299 | Robert Price | 09/2021 |
9899 | 26502^4297+4297^26502 | 96287 | Hans Havermann | 07/2020 |
9898 | (54180^20341-1)/54179 | 96287 | Serge Batalov | 06/2022 |
9900 | 2^319838+4257 | 96281 | Lelio R Paula | 03/2016 |
9901 | 2^319830-16995 | 96279 | Lelio R Paula | 03/2016 |
9902 | 6^123713+143 | 96268 | Lelio R Paula | 05/2016 |
9903 | (28*10^96265-97)/3 | 96266 | Robert Price | 07/2016 |
9904 | (3799^26893-1)/3798 | 96265 | Serge Batalov | 06/2022 |
9906 | 103^47825-102 | 96264 | Norbert Schneider | 10/2016 |
9905 | (54041^20341-1)/54040 | 96264 | Serge Batalov | 06/2022 |
9907 | (2^319777+2082)/2 | 96263 | Stefano Morozzi | 09/2020 |
9908 | (2^319771+3686)/2 | 96261 | Stefano Morozzi | 09/2020 |
9909 | 2^319713-81 | 96244 | Lelio R Paula | 11/2012 |
9910 | 2^319711+213413 | 96243 | Enzo Doria | 09/2023 |
9911 | (55459*2^319706+1)/20167 | 96242 | Lelio R Paula | 07/2014 |
9912 | 22834^16393+16393^22834 | 96238 | Hans Havermann | 05/2020 |
9913 | 8*10^96236+27 | 96237 | Robert Price | 09/2021 |
9914 | 12^89153-43 | 96213 | Lelio R Paula | 06/2016 |
9915 | (25048^21871-1)/25047 | 96202 | Serge Batalov | 06/2022 |
9917 | (2*10^96195-59)/3 | 96195 | Robert Price | 05/2017 |
9916 | 9*10^96194-77 | 96195 | Robert Price | 08/2020 |
9918 | 2^319545+147 | 96193 | Lelio R Paula | 11/2012 |
9919 | 23357^13128+13128^23357 | 96189 | Hans Havermann | 02/2020 |
9920 | (19838^22381-1)/19837 | 96179 | Serge Batalov | 06/2022 |
9921 | (221^41023-1)/220 | 96172 | Serge Batalov | 06/2022 |
9922 | 25952^5073+5073^25952 | 96160 | Hans Havermann | 12/2019 |
9923 | (50891^20431-1)/50890 | 96157 | Serge Batalov | 06/2022 |
9924 | (50^56597+1)/51 | 96155 | Robert Price | 03/2014 |
9925 | 22313^20374+20374^22313 | 96149 | Hans Havermann | 12/2019 |
9926 | 22279^20668+20668^22279 | 96141 | Hans Havermann | 11/2019 |
9927 | (41*10^96137-59)/9 | 96138 | Robert Price | 10/2017 |
9928 | 2^319353-110343 | 96135 | Enzo Doria | 05/2014 |
9929 | 11^92309-42 | 96130 | Lelio R Paula | 05/2016 |
9930 | (21675^22171-1)/21674 | 96129 | Serge Batalov | 06/2022 |
9931 | 14^83869-13^83869 | 96125 | Jean-Louis Charton | 01/2009 |
9932 | (7*10^96112+143)/3 | 96113 | Robert Price | 04/2016 |
9933 | 26731^3940+3940^26731 | 96112 | Hans Havermann | 07/2020 |
9934 | 22640^17583+17583^22640 | 96109 | Hans Havermann | 07/2020 |
9935 | (19178^22441-1)/19177 | 96107 | Serge Batalov | 06/2022 |
9936 | 2^319237+65835 | 96100 | Lelio R Paula | 03/2016 |
9937 | 2^319194-1841 | 96087 | Yves Minet | 03/2013 |
9938 | 23949^10280+10280^23949 | 96084 | Hans Havermann | 07/2020 |
9939 | (1927^29251-1)/1926 | 96083 | Serge Batalov | 06/2022 |
9940 | 20^73847-31 | 96078 | Lelio R Paula | 05/2016 |
9941 | (60229^20101-1)/60228 | 96075 | Serge Batalov | 06/2022 |
9942 | (28044^21601-1)/28043 | 96074 | Serge Batalov | 06/2022 |
9943 | 50^56543-49 | 96065 | Norbert Schneider | 09/2018 |
9944 | 2^319107-109555 | 96061 | Enzo Doria | 04/2014 |
9945 | (3*2^319114-1)/2627 | 96060 | Lelio R Paula | 02/2014 |
9946 | 2^319098-112157 | 96059 | Enzo Doria | 06/2014 |
9947 | 15^81673+94 | 96055 | Lelio R Paula | 07/2016 |
9948 | (85761^19471-1)/85760 | 96052 | Serge Batalov | 06/2022 |
9949 | 2^319074+1871749 | 96051 | Enzo Doria | 05/2016 |
9950 | 25670^5517+5517^25670 | 96050 | Hans Havermann | 06/2020 |
9951 | 6*10^96037+19 | 96038 | Robert Price | 11/2021 |
9952 | 2^319007-112111 | 96031 | Enzo Doria | 06/2014 |
9953 | (57987^20161-1)/57986 | 96029 | Serge Batalov | 06/2022 |
9955 | 2^318946-1421 | 96013 | Yves Minet | 01/2013 |
9954 | (68310^19861-1)/68309 | 96013 | Serge Batalov | 06/2022 |
9956 | (20294^22291-1)/20293 | 96012 | Serge Batalov | 06/2022 |
9957 | (304^38669-1)/303 | 96008 | Serge Batalov | 06/2022 |
9958 | (1840^29401-1)/1839 | 95986 | Serge Batalov | 06/2022 |
9959 | 22830^16003+16003^22830 | 95982 | Hans Havermann | 02/2020 |
9960 | 2^318834-110411 | 95979 | Enzo Doria | 05/2014 |
9961 | 2^318831-1965 | 95978 | Lelio R Paula | 03/2016 |
9962 | (5*10^95976+7)/3 | 95977 | Erik Branger | 03/2014 |
9963 | (12797^23369-1)/12796 | 95975 | Serge Batalov | 06/2022 |
9964 | (10534^23857-1)/10533 | 95963 | Serge Batalov | 06/2022 |
9965 | 22871^15696+15696^22871 | 95962 | Hans Havermann | 01/2020 |
9966 | (2^318760+858)/2 | 95957 | Stefano Morozzi | 02/2021 |
9967 | 23167^13860+13860^23167 | 95953 | Hans Havermann | 01/2020 |
9968 | (823^32911-1)/822 | 95946 | Serge Batalov | 06/2022 |
9970 | 5*3^201089-8 | 95945 | Lelio R Paula | 08/2014 |
9969 | (81783^19531-1)/81782 | 95945 | Serge Batalov | 06/2022 |
9971 | 2^318711-139 | 95942 | Lelio R Paula | 11/2012 |
9972 | 2^318635-1167 | 95919 | Lelio R Paula | 03/2016 |
9973 | 15^81555+16 | 95917 | Lelio R Paula | 07/2016 |
9974 | (37*10^95895+377)/9 | 95896 | Robert Price | 10/2017 |
9975 | (22699*2^318527+1)/21 | 95890 | Lelio R Paula | 07/2014 |
9976 | 45^57997+196072 | 95882 | Enzo Doria | 10/2023 |
9977 | 2^318496-97305 | 95877 | Lelio R Paula | 03/2016 |
9978 | 22129^21496+21496^22129 | 95871 | Hans Havermann | 07/2020 |
9980 | 2^318466+15015 | 95868 | Lelio R Paula | 03/2016 |
9979 | 24847^7216+7216^24847 | 95868 | Hans Havermann | 07/2020 |
9981 | 23706^11065+11065^23706 | 95866 | Hans Havermann | 07/2020 |
9982 | (2863^27733-1)/2862 | 95865 | Serge Batalov | 06/2022 |
9983 | 6^123187-173129 | 95859 | Mike Oakes | 04/2011 |
9984 | 2^318423-487 | 95855 | Yves Minet | 03/2012 |
9985 | 10^95848+443709 | 95849 | Patrick De Geest | 10/2022 |
9986 | 10^95848-5000000001 | 95848 | Robert Price | 09/2022 |
9987 | 10^95848-61971 | 95848 | Patrick De Geest | 10/2022 |
9988 | (2^318359+678)/2 | 95836 | Stefano Morozzi | 09/2020 |
9989 | (10223*2^318350+1)/1563 | 95834 | Lelio R Paula | 07/2014 |
9990 | (8060^24533-1)/8059 | 95831 | Serge Batalov | 06/2022 |
9991 | (9853^23993-1)/9852 | 95814 | Serge Batalov | 06/2022 |
9993 | (2^318267+2550)/2 | 95808 | Stefano Morozzi | 09/2020 |
9992 | (5048^25873-1)/5047 | 95808 | Serge Batalov | 06/2022 |
9995 | 4*10^95799+13 | 95800 | Robert Price | 02/2021 |
9994 | (65533^19891-1)/65532 | 95800 | Serge Batalov | 06/2022 |
9996 | 2^318203+211343 | 95789 | Enzo Doria | 08/2023 |
9997 | 45369!2+1024 | 95787 | Robert Price | 02/2015 |
9998 | 3^200758-86 | 95786 | Lelio R Paula | 05/2016 |
9999 | (2^318271+1)/(3*46031404238891*654090604244843) | 95781 | Gord Palameta | 12/2018 |
10000 | (2^318171+2030)/2 | 95779 | Stefano Morozzi | 09/2020 |